1. The problem statement, all variables and given/known data For n = 1, 2, ..., let fn be a Lebesgue integrable function [0,1] → [0, +∞) such that (1) ∫01 fn dx = 1 and (2) ∫1/n1 fn dx < 1/n Let g(x) = supn ∈ ℕfn(x). Prove ∫01 g(x)dx = +∞ 3. The attempt at a solution Coffee, banging my head against a wall, etc. There's not enough information about the behaviour of the fn themselves, apart from the integrals, to use any of the convergence theorems. Fatou's lemma seems like an obvious choice since it only deals in integrals, but choosing a positive and measurable function that leads to any results is daunting. This is another one of those problems where I get why the answer is what it is on a non-rigorous level but have no idea how to apply theory ... I thought about turning the integral of g into an infinite sum of integrals from 1/(n-1) to 1/n, but since we can concentrate the support of fn in [0, 1/n] on as small an interval containing 0 as we want, we have no guarantee that any of those individual integrals is anything but positive. Aaaaargh.